Theorem 1 (Noncommutative Freiman-Kneser theorem for small doubling) Let , and let be a finite non-empty subset of a multiplicative group such that for some finite set of cardinality at least , where is the product set of and . Then there exists a finite subgroup of with cardinality , such that is covered by at most right-cosets of , where depend only on .

One can of course specialise here to the case , and view this theorem as a classification of those sets of doubling constant at most .

In fact Hamidoune’s argument, which is completely elementary, gives the very nice explicit constants and , which are essentially optimal except for factors of (as can be seen by considering an arithmetic progression in an additive group). This result was also independently established (in the case) by Tom Sanders (unpublished) by a more Fourier-analytic method, in particular drawing on Sanders’ deep results on the Wiener algebra on arbitrary non-commutative groups .

This type of result had previously been known when was less than the golden ratio , as first observed by Freiman; see my previous blog post for more discussion.

Theorem 1 is not, strictly speaking, contained in Hamidoune’s paper, but can be extracted from his arguments, which share some similarity with the recent simple proof of the Ruzsa-Plünnecke inequality by Petridis (as discussed by Tim Gowers here), and this is what I would like to do below the fold. I also include (with permission) Sanders’ unpublished argument, which proceeds instead by Fourier-analytic methods. Read the rest of this entry »

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